Mary Rees

The geometric model direct from Teichmuller geodesics.

Abstract: Minsky's proof of the Ending Laminations Conjecture for any Kleinian surface group $\Gamma $ can be considered to consist of three stages:

  1. the construction of a geometric model up to quasi-isometry, given the ending laminations of $H^{3}/\Gamma $;

  2. the constrction of a map from the geometric model to $H^{3}/\Gamma $ which is Lipschitz, at least onto the thick part of $H^{3}/\Gamma $;

  3. a proof (with Brock and Canary) that this map is, in fact, biLipschitz.

The extant construction of the geometric model and the Lipschitz map uses the curve complex and deep and extensive work of Masur and Minsky on hierarchies of tight geodesics in the curve complex. The purpose of the present talk will be to show that there is a natural construction of the geometric model using Teichmuller geodesics directly. I shall describe briefly a number of results about Teichmuller geodesics, which are used in this direct construction of the geometric model, and which suggest that a direct construction of a Lipschitz map to the thick part of $H^{3}/\Gamma $ should be possible.

Background material on this topic can be found in chapters of two preprints on my web-page at www.liv.ac.uk/maths: Chapters 14 and 15 of ``Views of Paramter Space: Topographer and Resident'' (and also Chapter 28, to a lesser extent) and in ``Notes on Geometry in Hyperbolic and Teichmuller Space''

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